fabio rodrigo siqueira batista , tara keshar nanda baidya ...fabio rodrigo s. batista, tara k.n....

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Pesquisa Operacional (2013) 33(3): 361-382© 2013 Brazilian Operations Research SocietyPrinted version ISSN 0101-7438 / Online version ISSN 1678-5142www.scielo.br/pope

DETERMINATION OF THE CARBON MARKET INCREMENTAL PAYOFFCONSIDERING A STOCHASTIC JUMP-DIFFUSION PROCESS

Fabio Rodrigo Siqueira Batista1*, Tara Keshar Nanda Baidya2,Jose Paulo Teixeira3 and Albert Cordeiro Geber Melo4

Received December 15, 2011 / Accepted March 15, 2013

ABSTRACT. The objective of this paper is to verify the robustness of the Least Square Monte Carlo and

Grant, Vora & Weeks methods when used to determine the incremental payoff of the carbon market for

renewable electricity generation projects, considering that the behavior of the price of Certified Emission

Reductions, otherwise known as Carbon Credits, may be modeled using a jump-diffusion process. In addi-

tion, this paper analyses particular characteristics, such as absence of monotonicity, found in trigger curves

obtained through use of the Grant, Vora & Weeks method to valuate these types of project.

Keywords: least square Monte Carlo, grant, Vora & Weeks, jump-diffusion process, carbon market,

renewable sources of energy.

1 INTRODUCTION

The enforcement of the Kyoto Protocol and the fines imposed upon European corporationsthat do not manage to reduce their Greenhouse Gases (GHGs) emissions is making the carbon

market a reality in Latin America. According to studies carried out by the Brazilian Government(see Brazil, 2005), Brazil stands out as one of the countries of highest potential in the export ofCertified Emission Reductions (CERs), with the capacity to reduce its annual emissions by up to

an equivalent of 120.6 million tons of carbon dioxide. It is worth highlighting that a large part ofthis potential is due to its capacity to produce electrical energy from renewable resources.

As established by the UNFCCC (1998), any project that is developed following the rules es-tablished by the Clean Development Mechanism under the Kyoto Protocol, also called CDM

*Corresponding author1Professor Adjunto, Pontifıcia Universidade Catolica do Rio de Janeiro – PUC/Rio. Pesquisador, Centro de Pesquisasde Energia Eletrica – CEPEL, Brazil. E-mail: [email protected] Associado, Universidade do Grande Rio – UNIGRANRIO/ECT. E-mail: [email protected] Associado, Pontifıcia Universidade Catolica do Rio de Janeiro – PUC/Rio. E-mail: [email protected] Geral, Centro de Pesquisas de Energia Eletrica – CEPEL. Professor Associado, Universidade Estadual do Riode Janeiro – UERJ. E-mail: [email protected]

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362 DETERMINATION OF THE CARBON MARKET INCREMENTAL PAYOFF

projects, must have its additionality proven before being registered as such by the United Na-

tions Executive Board. This means that, amongst other things, it must be proven that the project’sGHG emissions are lower than those for the baseline scenario, which can be defined as the emis-sion scenario that would be observed if the proposed project were not implemented. Therefore,

motivated by the strong conservatism adopted by the Executive Board in recognizing the addi-tionality of proposed projects, different methodologies have been developed to determine a re-liable baseline, among which is the ACM0002 methodology – Approved Consolidated Method-

ology number 2 (see UNFCCC, 2006), which also makes reference to the Tool to calculate theemission factor for an electricity system (see UNFCCC, 2009), that is used in this paper.

According to the scope of the ACM0002 methodology, its objective is to determine the baselinefor CDM renewable electricity generation projects, where it is required that they are connected

to the electrical grid of the country in which the project will be implemented. In this case it isworth noting that the baseline becomes a direct function of the operation of all of the powerplants connected to the grid in which the project will be carried out.

With this in mind, Batista et al. (2011a) proposed a methodology capable of determining the in-

cremental payoff of the carbon market for grid-connected renewable electricity generation CDMprojects, considering that its baseline behaves as a random variable. In this case, the built-inrandomness of the baseline is intended to account for uncertainties associated with the operation

of the electricity system in which the CDM project will be connected, in this way including arisk factor in calculating the quantity of CERs that the project will produce and commercializein the future.

In addition, Batista et al. (2011a) considered the uncertainty of the market due to the randomness

in CER price when determining the incremental payoff of the carbon market. For this purpose, itwas considered that renewable electricity generation projects may have sufficient flexibility to beregistered by the United Nations Executive Board, thus allowing the commercialization of CERs

generated through their operation. Such flexibility can be understood as an American option, andin this paper it is evaluated as such.

In this analysis the binomial method (see Cox, Ross & Rubistein, 1979) was initially used toevaluate this previously mentioned flexibility. Then the Least Square Monte Carlo (see Longstaff

& Schwartz, 2001) and Grant, Vora & Weeks (1996) methods were also used by Batista et al.(2011b). Despite the fact that a convergence of results was observed for all of the methods used,the LSM and GVW methods are considered to be more flexible than the binomial method, sincethey can be used for the evaluation of different types of options or for options with different

stochastic factors.

In this context, the main contribution of this paper is to continue the analyses carried out byBatista et al. (2011b) verifying the robustness of the Least Square Monte Carlo (LSM) and Grant,Vora & Weeks (GVW) methods when the behavior of the CER price can be modeled using a

jump-diffusion process, instead of the traditional geometric Brownian motion used by Batistaet al. (2011b). As written by Merton (1976), it is known that in the presence of jumps, the

Pesquisa Operacional, Vol. 33(3), 2013

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FABIO RODRIGO S. BATISTA, TARA K.N. BAIDYA, JOSE P. TEIXEIRA and ALBERT C. GEBER MELO 363

Black-Scholes “hedge” portfolio will not be riskless, and hence, their ‘no arbitrage’ technique

cannot be employed in order to derive an analytical formula for option pricing. Even if one knewthe required expected return on the option, the resultant mixed partial differential-differenceequation would be difficult to solve. Based on these statements, we decided to resort to the use

of numerical methods for option valuation, which justifies the above analysis.

In addition, another contribution of this paper is considered to be the analysis of particular char-acteristics of the trigger curves obtained through use of the GVW method for CDM projects valu-ation. Among other characteristics, these curves do not present the generally observed monotonic

format found, for example, in the classic works of Siegel, Smith & Paddock (1987) and Dixit &Pindyck (1994).

2 JUMP-DIFFUSION PROCESS

In the international carbon market it can be observed that CER price estimates have a high in-

herent randomness, owing to existent uncertainties in future supply and demand. For example,between April 24 and May 12 in 2006, the finding that European Union Allowances had beendelivered to European corporations in excess caused the drop in the price of these assets from

29.43 to 10.14 e/tCO2e. More recently the occurrence of jumps on the CER price can also be ob-served between October 14 and October 27 in 2008, when the price dropped from 20,26e/tCO2eto 15,17 e/tCO2e; between February 2 and February 12 in 2009, when the price dropped from

10,54 e/tCO2e to 7,6 e/tCO2e; and between July 26 and August 8 in 2011, when the pricedropped from 9,99 e/tCO2e to 7,75 e/tCO2e (see http://www.bluenext.fr/statistics/graphs.html).

Taking into account the inadequacy of the standard stochastic diffusion model in representing thereality generally observed in the market, or that the financial and non-financial asset values can

present slight random jumps in response to the arrival of new information, alternative stochasticmodels have been developed. In this paper, we highlight the use of the jump-diffusion model.

The jump-diffusion model was initially proposed by Merton (1976) for the pricing of financialoptions. In order to adequately model the behavior of financial assets, Merton considered that

the irregularity in the price of these assets must be composed of two types of vibrations: normaland abnormal.

According to Merton (1976), normal vibrations can occur due to a temporary imbalance in thesupply and demand of the asset, caused by changes in the economic scenario or simply by the

arrival of new information capable of causing marginal changes in its price. Merton suggests thatthese vibrations may be modeled using the geometric Brownian motion.

On the other hand, abnormal vibrations are considered the consequence of new information capa-ble of producing a larger than marginal effect on asset prices. Generally, such information is spe-

cific to a particular company or industry. Merton suggests that this type of vibration may be mod-eled using a Poisson process. It is worth highlighting that in this model the Poisson distributionevent is the arrival of new information capable of producing jumps in asset price. These events


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are considered independent and similarly distributed. Equation 1 describes the jump-diffusion

model proposed by Merton:

d S

S= (α − λk)dt + σdz + dq . (1)

In equation 1, S represents the price of the financial asset, α represents the instantaneous expected

return of the asset, σ 2 represents the instantaneous variance in the return if the Poisson event doesnot occur, dz is the increment of the Wiener process, dq is the increment of the Poisson process,λ represents the average number of Poisson event arrivals per unit of time, and k representsthe expected percentage variation (Y − 1) of the asset price if the Poisson event occurs, or,

k = E(Y − 1), where Y is an impulse function producing a finite jump in S to SY . Note that inthis model dz and dq are independent processes.

Supposing that in a small interval of time dt there exists a 100% chance that a maximum ofone jump will occur, the previously mentioned stochastic process can be written in the following

way:d S

S= (α − λk)dt + σdz + (Y − 1) . (2)

So, for the opposite situation, where there is a 100% chance that no jump will occur within timeinterval dt , it must follow that:

d S

S= (α − λk)dt + σdz . (3)

In other words, it can be deduced from equations 2 and 3 that in the absence of jumps the assetprice will follow the same dynamics as the model proposed by Black & Scholes (1973), exceptfor the presence of factor λk in the process trend term.

Note that λk must be introduced into equation 1 to correct a potential bias introduced by the useof the Poisson process. As shown in equation 4, this bias owes itself to the fact that the expectedvalue of the Poisson process is not zero:

E(dq) = λdt E(Y − 1) + (1 − λdt) · 0 = λdtk . (4)

Therefore, for the expected return of an asset that follows the jump-diffusion process to be thesame as that obtained when the standard diffusion process is used, it is necessary that the processtrend term be subtracted from E(dq).

In this paper the jump-diffusion process described in equation 1 is used to model the movement

of the CER price. Solving this stochastic differential equation, we obtain the dynamics of theasset price as (see Tsay, 2002):

St = S0 exp

[(α − λk − σ 2

2

)t + σ zt

]Y (n) . (5)


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In equation 5, zt represents a random variable with normal distribution, with zero mean and

variance t . Still in equation 5, Y (n) = 1 if n is equal to zero, and Y (n) = ∏nj=1 Y j if n ≥ 1,

where Y j are independently and identically distributed, (Y j − 1) represents the percentage vari-ation in the stock price if the jth poisson event occurs, and n is the number of occurrences of

the Poisson event distributed within a time interval [0, t ]. In this paper the price paths ob-tained from equation 5 will be used in the GVW and LSM methods for the valuation of theconsidered option.

As described in Section 1, the contributions of this paper are: the proof of the robustness of

the GVW and LSM methods in the pricing of the carbon market incremental payoff for someelectricity generation projects when the CER price follows a jump-diffusion process, and theanalysis of particular characteristics of the trigger curves obtained through the use of the GVW

method. In both cases, it is considered that full understanding of the results requires that thereader understand the baseline calculation method used in this paper. This will be discussed inSection 3.

3 BASELINE CALCULATION IN ELECTRICAL INTERCONNECTED SYSTEMS

According to the ACM0002 methodology, the baseline of grid-connected renewable electricitygeneration projects should be determined using a combination of two types of emission factor:the Operating Margin Emission Factor (EFOM) and the Build Margin Emission Factor (EFBM),

as described in equation 6.

E Fy = 50% · E FO M,y + 50% · E FBM,y . (6)

In this equation, note that y represents the year for which the baseline is being calculated. Regard-ing the determination of E FO M , the ACM0002 methodology establishes four different methodsthat can be applied to grid-connected generation projects: the Simple, Simple adjusted, DispatchData Analysis, and the Average Method. Given that the objective of this paper is to verify the

robustness of different numerical methods in the valuation of the considered option, and that acomparative analysis of the different methods for baseline emission factor calculations has al-ready been carried out by Batista et al. (2011a), in this paper only the Average method will be

used. This decision is justified through its simplicity when used in Operating Margin EmissionFactor calculations.

According to the methodology adopted by the Average Method, the Operating Margin EmissionFactor calculation must be carried out using the following equation:

E FO M,y =∑

n GE N j,y · C O E Fj∑n GE N j.y

(7)

where GE N j,y represents the quantity of electrical energy (in MWh) produced by power plant

j in year y, C O E Fj represents the carbon dioxide emission coefficient (in tCO2/MWh) of theprimary energy source used by power plant j , and n represents the total number of power plantsthat belong to the grid where the CDM project is located.


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Once the Operating Margin Emission Factor has been determined, the Build Margin Emission

Factor must be calculated based on the largest annual generation determined from the followinggroups of power plants:

• the last five power plants that have been built within the grid where the CDM project is

located;

• the most recent additions to the grid capacity of the project, which represent 20% of itstotal generation.

In both cases, all previously built CDM projects must be excluded from calculation of the BuildMargin Emission Factor, which can be determined using equation 7. In this case, parameter nrepresents the set of power plants as defined by one of the two groups just mentioned. Once

E FO M and E FBM have been determined, equation 6 can be used to determine the CDM projectbaseline emission factor.

It is worth highlighting that the calculated baseline emission factor takes into account that arenewable electricity generation project reduces carbon dioxide emissions by substituting the

energy produced by power plants that burn fossil fuels and are connected to the same grid ofthe project. Therefore, the CDM project baseline becomes a direct function of the operation ofall power plants within the system, since the greater the generation due to thermoelectric plants,

which in general use fossil fuels extensively, the greater will be the baseline for the CDM projectbeing considered.

Emission reductions achieved by the CDM project can be calculated mathematically in the fol-lowing way:

REy = E By − E PY − Fy (8)

where RE represents the reduction in carbon dioxide emissions achieved through operation ofthe CDM project, E B represents the emissions corresponding to the baseline, E P represents the

emissions of the CDM project itself, and F represents any leakages or indirect project emissions.Also in equation 8, note that y represents a period of one year over which project activity ismonitored in order to account for the reduction in carbon dioxide emissions.

It is important to say that, for renewable electricity generation projects, such as small hydroelec-

tric plants or wind farms, the ACM0002 methodology determines that both the emissions and theleakages of a CDM project must be considered to be zero. As a consequence, baseline emissionsmust be determined in the following way:

E By = EGy · E Fy (9)

where EGy represents the electricity generation of the CDM project and E Fy represents itsbaseline emission factor, both determined for a given year y. In this paper it is considered that

CDM project generation can be calculated from the product of its installed capacity and thecapacity factor of the plant.


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Based on the details of the ACM0002 methodology, we concluded that calculating the time

evolution of the baseline of such generation projects involves prior recognition of the followingvariables throughout the period of operation: (1) the thermal and hydraulic generation withinthe grid; (2) the primary resource of the energy used for each operation within the grid; and,

finally, (3) the configuration of the grid’s thermal and hydraulic expansion, the latter being ofgreat importance in the calculation of E FBM .

Finally, the estimation of these parameters is important in that it enables the estimation of futurebaseline scenarios and contributes towards the accuracy of economic-financial viability analyses

of CDM projects, the results of which will aid investor decision making.

4 PROPOSED METHODOLOGICAL APPROACH

The objective of this section is to provide a general overview of the solution process adoptedin this paper to determine the incremental payoff of the carbon market for renewable electricity

generation projects using the GVW and LSM methods. Figure 1 illustrates the proposed method-ological approach.

GVW METHOD LSM METHOD

GENERATION OF BASELINE EMISSION FACTOR SCENARIOS

DETERMINATION OF THE CARBON MARKET INCREMENTAL PAYOFF

PROJECT DATA

GENERATION OF HYDROLOGICAL DISPATCH SCENARIOS

GENERATION OF OPERATING MARGIN EMISSION FACTOR SCENARIOS

GENERATION OF BUILDING MARGIN EMISSION FACTOR SCENARIOS

Figure 1 – Proposed methodological approach.

Once the technical and economic characteristics of the CDM project and of the grid are known,the first step is to determine, over time, several hypothetical dispatch scenarios over time for the

power plants that belong to the grid configuration.


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Assuming that such hydrological scenarios may be determined, the next two steps are to esti-mate, for each previously defined scenario, the Operating and Build Margin emission factors ofthe grid. Such factors are determined according to the criteria established within the ACM0002methodology. After the determination of these emission factors, the next step is to determinescenarios for the CDM project baseline emission factor. To do this equation 6 is used. Oncethis stage is complete, it is considered that the risk related to the total amount of CERs that willbe produced by the project is found to be adequately represented by the scenarios that modelthe evolution of its baseline.

The final step of the process involves determining the incremental payoff of the carbon marketfor the CDM project. Once the investor has the right, but not the obligation, to make the additionalinvestment of registering their project with the United Nations Executive Board, it is consideredthat the project holder has a management option. Such an option allows the investor to apply forand trade CERs during the operational phase of the project. In this paper it is considered thatthe GVW and LSM methods can be used to valuate this option, and that the CER price can bemodeled using a jump-diffusion process.

It is also important to observe that the value of the option is determined for each of the previouslyestimated baseline scenarios, and that the average of these values represents the final value of theevaluated option.

As explained in Section 3, in order to calculate E FO M and E FBM it is necessary to know whichprimary energy resources will be used by each power plant within the grid, as well as the con-figuration of its thermal and hydraulic expansion. In this paper, since we will consider that theCDM project will be developed in the Brazilian Interconnected System, the platform data usedby the Brazilian Ministry of Mining and Energy in carrying out its Ten Year Electrical EnergyExpansion Plan – PDE (see Brazil, 2006) will be used in our analyses. It is important to say thatthe PDE analyses are driven by long-term Brazilian electrical energy system guidelines, thesebeing responsible for the identification of main lines of development within electrical energygeneration and transmission systems in Brazil.

Note that, in order to obtain thermal and hydraulic energy generation scenarios, which are alsorequired for the calculation of E FO M and E FBM over time, a long-term operation planningmodel is also needed. In this paper, these information will be provided by the Newave model(see Maceira, 2008), which will use the PDE platform data in order to carry out energy operationplanning for the Brazilian Interconnected System. The results are obtained monthly for a periodof up to ten years.

It should be noted that the Newave model uses Stochastic Dual Dynamic Programming (seePereira, 1991) as a solution methodology, as well as being officially used by the System Operatorto carry out energy operation planning in Brazil.

Finally, it is important to highlight that the methodology used in this paper comprises someparticular features that are not usually found in other papers that provide economic evaluation ofrenewable electricity generation CDM projetcs. These features are: (i) the random nature of thebaseline scenario; and (ii) the random nature of the CER price.


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As an example of related works, Roques et al. (2006) uses the Real Options Approach to de-termine the real value of a Nuclear Power Plant when compared to other fossil fuel genera-tion technologies in Europe, thus considering that the gas and the carbon prices are stochastic.In addition to this, Reedman et al. (2006) also uses the Real Options Approach to model elec-tricity generation technology adoption in Australia under the carbon price uncertainty, thus con-sidering that the investment decision can be driven by the introduction of a carbon penalty tomajor emitters.

Concerning to the estimation of baseline scenarios, most works are concentrated on land useprojects. For example, Dale et al. (2003) projects the effects that land use may have upon land-cover change and carbon storage in the Eastern Panama Canal Watershed. In this case, a spatialmodeling approach is used in order to determine regional-scale baseline emissions scenarios.Jong et al. (2005) also used a spatial modeling approach for testing and applying a regionalbaseline for carbon emissions from land-use change in Mexico. None of these works considerthe stochastic behavior of the baseline scenario.

5 CASE STUDY

The results and conclusions described in Sections 6 and 7 of this paper are based on the applica-tion of the proposed methodology on a wind energy generation project with 135 MW of installedcapacity being implemented in the southeast region of Brazil. It is considered that this project is

connected to the Brazilian Interconnected System and subject to all terms of the local legislation,such as taxes and industry charges applied in Brazil.

In this case study, transaction costs of US$ 137,500.00 are assumed. These costs represent theadditional investment necessary so that all stages of project registration process can be completed

with the United Nations Executive Board. It is assumed that the registration process will becomplete twelve months after the additional investment has been made, this being the averagetime taken for the registration of Brazilian projects. Note that only after its registration can the

candidate project be classified as a CDM project, thereafter having its future emission reductionsconverted into CERs.

In the operational phase of the project, annual costs of US$ 9,000.00 are assumed for the ver-ification and certification of CERs, as well as 11% of its total value for issuing and trading.

A 43.25% tax payment on revenues gained from the annual sale of CERs was also assumed. Itwas considered that this revenue should be calculated in the following way:

R(y) = P(ι) · REy · (1 − C EC) − C F (10)

where R(y) represents liquid revenues for year y, P(ι) represents the CER price at time (ι)

in which the investment option was exercised, REy represents the quantity of CERs generatedby the project for year y (see Eq. 8), C EC represents the percentage of expenditures involved

in CER issuing and trading, and C F represents the annual costs due to CER verification andcertification.


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Note that the period taken into account for the attainment of CERs is 10 years from the imple-

mentation of the project, or in other words, only the CERs obtained during this period will begranted to the project and may then be traded. This assumption is in line with the regulationsimposed by the international carbon market.

It should then be clear that in following these adopted assumptions, the decision to register the

wind energy project can be made up to twenty-four months after the start of its construction,thereby defining the maturity of the considered investment option. Note that within this periodthe optimum moment to exercise the option and register the project with the Executive Board

is guided by the price level that the CERs might achieve on the international carbon market. Tothis end, it is considered that the price dynamic follows a jump-diffusion process, with its pathbeing simulated using equation 5. It is assumed that the observed jumps in CER price follow

a normal distribution, with an average of 30% and a standard deviation of 15% for the currentCER value. In addition, an average occurrence of two jumps per year is assumed for the simula-tion of CER prices.

All previously outlined assumptions, together with all other considered assumptions, are de-

scribed in Table 1.

Table 1 – Information needed for the general case study.

Description Unit Value

Installed Capacity MW 135,00

Capacity Factor % 23,49Plant Construction Period months 30

Annual Cost of Capital % 12,00Annual Risk-Free Interest Rate % 8,00

Total Registration Period months 12CER Verification/Certification Costs US$/Year 9.000,00

CER Issuing/Trading Costs % 11,00CER Attainment Period years 10

Exchange Rate R$/US$ 2,20

Global Taxes % 43,25

Table 2 – Parameters needed for the option pricing under the CER process.

Description Unit Value

CDM Total Registration Costs US$ 137.500,00Initial CER Price US$/tCO2e 5,00

Annual Volatility of CER Price % 40,00Annual Dividend Yield % 5,00

Option Maturity months 24Annual Average Occurance of Jumps – 2

Average Jump Size % CER Price 30,00Jump Size Standard Deviation % CER Price 15,00


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6 RESULTS

In this section we will present the results of the investigation into the robustness of the LSM andGVW methods when used to determine the incremental payoff of the carbon market for grid-connected electricity generation projects. As well as this, specific characteristics of the triggercurves obtained using the GVW method will also be analyzed.

Initially, it is worth noting that the robustness of these methods has already been tested by Batistaet al. (2011b) for conditions similar to those described in this paper, but considering CER pricemodeled using a geometric Brownian motion (GBM). In their study the results obtained usingthe GVW and LSM methods were compared with the option value obtained using the binomialmethod proposed by Cox, Ross & Rubinstein (1979), the result of which was considered to bea benchmark in the analyses performed. In this paper the convergence of the GVW and LSMmethods was tested assuming that the CER price dynamic follows a jump-diffusion process. Asdescribed in Section 1, it is known that in the presence of jumps the no-arbitrage assumptionis no longer valid, which virtually prevents the pricing of American options through analyticalsolutions, requiring the use of numerical methods to do so.

Considering that, in the presence of jumps the valuation of options using the binomial methodalso becomes rather complex, so the results obtained from the GVW method were used as abenchmark in our analyses. Therefore, 100,000 simulated price paths were used in each iterationof this method. Additionally, 96 early exercise dates were used, which is equivalent to dividingthe option lifetime (twenty-four months) into weekly time periods. This choice was based on theresults obtained by Batista et al. (2011b), which in the absence of jumps show that good optionvalue estimates can be obtained using 40,000 simulated paths and 48 early exercise dates.

The measure of accuracy used in these analyses was the percentage Root Mean Square Error(RMSE) of the estimator, whose formula is described in equation 11:

CV (C) = RM S E(C)

CRE F× 100

√V (C) + [C − CRE F ]2

CRE F× 100 (11)

where CV represents the coefficient of variation or the percentage RM S E of the estimator C ,CRE F represents the true value of the option, which will be estimated using the GVW method,and V (C) represents the variance of estimator C , which will be determined using the LSMmethod.

One of the main characteristics of the LSM method is that it supposes that the continuationfunction of the option can be represented by a linear combination of base functions. Note thatthe continuation function can be defined as the function that, at any time, provides the valueof waiting for new information and decide in the future if it is worth exercising the option.According to Longstaff & Schwartz (2001), various types of functions can be used for this, forexample, the Laguerre, Legendre, Chebyshev or Jacobi polynomials.

In this paper the same type of base function was used as that used by Longstaff & Schwartz intheir original article:

B1(S) = Sl , 1 = 1, 2, 3, � (12)


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where S represents the price of the underlying asset and l represents the term of the continua-tion function corresponding to the respective base function. As shown in equation 13, it shouldbe highlighted that Longstaff & Schwartz use the linear combination of two base functions toestimate the continuation value of the option:

FG(w, t) =G∑

l=0

al Bl(S) =2∑

l=0

al Bl (S) = a0 + a1 · S + a2 · S2 (13)

Note that the continuation value of the option, when compared to its immediate exercise value,determines, at any time, if it is optimum to exercise the option, given a simulated price path anda given baseline scenario.

As previously described, the methodology proposed in this paper uses various hydrological dis-patch scenarios in order to model the uncertainty associated with the CDM project baseline.First, the option is priced for each hydrological scenario, then its final value is obtained from theaverage of the values obtained in each scenario.

Since the GVW method is intensive in the use of Monte Carlo Simulation, the computationalcost of this method is fairly high when compared to the binomial or the LSM method. Thisproblem is further aggravated as the number of baseline scenarios is higher, therefore, for nbaseline scenarios, the algorithm proposed by GVW (see Appendix A) should be repeated ntimes. Because of this, only the first ten hydrological dispatch scenarios generated by the Newavemodel were used in the convergence analyses.

In the convergence analyses, the variation coefficient related to the least accurate estimativeamong all baseline scenarios was used. In other words, from all ten baseline scenarios considered,the result used for the convergence analysis is the largest variation coefficient that was found.It was considered in this paper that the estimators with variation coefficients lower than 5%already provide a good approximation of the real option value. The results found are presentedin Table 3.

Table 3 – Accuracy of the LSM method.

No. of Number of early exercise dates

simulated paths 12 24 48 96

5.000 7,89% 9,19% 6,40% 6,25%10.000 7,10% 6,81% 4,04% 3,61%

20.000 7,12% 4,38% 4,09% 2,74%40.000 6,26% 4,89% 2,72% 3,40%

100.000 6,34% 3,03% 2,50% 1,59%

Analyzing the results in Table 3 for 100,000 simulated paths and 96 early exercise dates, weconcluded that, among all of the considered baseline scenarios, the estimate of lowest accuracypresented a CV value equal to 1.59%, or in other words, the estimates associated with all otherscenarios show CV to be lower than this value. In addition, as the number of simulated paths


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and/or the number of expected option exercise dates increases, the option values calculated bythe LSM method tend to converge with the reference value as determined by the GVW method.

From these results, it can be concluded that even when the CER price follows the jump-diffusionprocess, the LSM and GVW methods can be considered robust in the determination of the carbonmarket incremental payoff.

Within the particular characteristics of the option evaluated by this study, it is important to notethe relationship between the value of the underlying asset (the present value of revenue flowsobtained through CER sales), the simulated price for the CER itself, and the considered baselinescenario. For example, considering an investment option (C) under the present value (V ) previ-ously mentioned, whose value is a function of CER price (S), it is possible to say that there islinearity between values V and S when S is modeled using some specific stochastic processes.For a given instant, supposing that:

V = Q · S (14)

where Q represents the number of CERs produced by the project, and S represents the CERprice modeled by a jump-diffusion process, it can be deduced that:

d S = (α − λk)Sdt + σ Sdz + Sdq (15)

d(Q · S) = (α − λk)QSdt + σ QSdz + QSdq (16)

dV = (α − λk)V dt + σV dz + V dq (17)

or, that the value of project V follows the same stochastic process followed by the CER price.Note that this conclusion may have a direct influence on the GVW method solution process.Since the trigger curve of an American option depends only on the dynamics of the value of itsunderlying asset (V ) (see Appendix A), which, in this case, is determined from a random price(S) and from a deterministic Q, it is initially expected that a single trigger curve might be validfor all baseline scenarios.

In addition, note that the trigger curves are generally characterized as monotonic functions (seethe work of Siegel, Smith & Paddock, 1987, and Dixit & Pindyck, 1994). For example, it isexpected that trigger curves for American options may be strictly decreasing functions of time.According to Hull (1993), this occurs because the closer you get to the expiry date, the lower theopportunity cost for its early exercise, which, as a consequence, reduces its value.

In an attempt to verify the previously stated intuition, trigger curves obtained through use of theGVW method were plotted for two distinct baseline scenarios. The results are shown in Figure 2.

The results in Figure 2 show that the trigger curves do not present a monotonic format, andfurthermore, that these curves are hardly equivalent to one another, varying according to thebaseline scenario considered. Both results contradict the previously described intuition, and canbe explained through analysis of parameter Q described in equation 14.

For the problem in question, Q represents the quantity of CERs produced by the project, whichdepends as much on the wind energy project generation, considered to be a constant in this


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Figure 2 – Comparison of trigger curves of different baseline scenarios.

study, as it does on the baseline emission factor. Since the baseline emission factor is updated ona monthly basis, it can be concluded that Q, despite being able to be considered deterministic atthe moment in which the option is valued, is not constant over time. This consideration directlyeffects the relationship between the probability distributions of V and S, which are then no longerconstants.

Note that this finding is independent of the stochastic process considered. For example, consid-ering that S follows an GBM such as that described in equation 3, note that the following resultcan be obtained:

Vt+�t = Qt+�t · St+�t (18)

It is known that at t + �t the expected value of a random value that follows an GBM is given by(see Hull, 1993):

E[St+�t

] = St · er.�t (19)

Therefore, it follows that:

E[Vt+�t

] = Qt+�t · St · er.�t = Qt+�t · E[St+�t

](20)

Using the same rational for the variance of the project value, it follows that:

V ar[Vt+�t

] = Q2t+�t · S2

t · e2.r�t · [eσ 2�t − 1] = Q2

t+�t · V ar[St+�t

](21)


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In summary, it can be concluded that if the relationship between the probability distributions forV and S is constant, the trigger curve for the derivative can be characterized as a function thatis monotonic and is independent of the baseline scenario. However, in the opposite case noth-ing can be said about its monotonicity or independence from possible non-financial uncertaintyscenarios. The determination of trigger curves with the characteristics described in this paperis original, diverging from what has been seen by Siegel, Smith & Paddock (1987) and Dixit &Pindyck (1994), for example.

Assuming that the baseline emission factor of the project is constant over time, note that themonotonicity of the trigger curves, as much as their independence in relation to the non-financialuncertainty scenario tend to be re-established. These results are illustrated in Figure 3.

Figure 3 – Comparison of trigger curves for different baseline scenarios with constant emission factorsover time.

Back in Figure 2, note that where instant t is equal to one year, it was not possible to find thetrigger curve value for baseline scenario 2. This means that regardless of the price that the CERreaches, early exercise of the option would not be optimum at this instant. Such a fact may alsobe attributed to the variation of the CDM project baseline during the study period.

In Figures 2 and 3 it is also shown that exercise of the option is not optimum in the first sixmonths of its lifetime. This is explained in noting that once the option is exercised, twelve monthsare necessary for the project to be registered by the Executive Board. In addition, even once theproject is registered, CERs will only start to be produced when the project is in operation, orwhen the construction of the power plant is complete. Since the time period for construction of


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the plant is eighteen months, it is clear that exercise of the option during the first six months ofconstruction would generate a financial cost due to a larger time interval between the additionalinvestment and the start of CER production.

The results presented until now consider a CER price modeled by a jump-diffusion processwhereby the jumps may be either positive or negative, or, in other words, when a random jumpoccurs, CER price can either rise or fall. Considering only the occurrence of negative jumps,thus reducing the initial value of the CER, it is noted that the trigger curves present different be-havior to that previously described. Figure 4 shows the trigger curves determined using differentaverage values (μ�) for random negative jumps.

Figure 4 – Trigger curves assuming only negative jumps in the CER price paths.

Contrary to what was observed in Figure 3, the use of a jump-diffusion process for exclusivelynegative jumps results in the possibility of early exercise of the option being optimum, even

during the first six months. In this case, it is noted that for large CER price levels (or large projectvalues), the opportunity cost for non-immediate exercise is high. This is because in the futurethe CER price would be subject to sudden reductions in value. In this case, it is noted that the

immediate exercise value of the option may surpass the benefit of waiting for new information,even if there is a financial cost associated with a larger gap between the registration date ofthe project and the start of its operation. Upon analyzing the situation where both positive and

negative jumps can occur (Fig. 2), note that such an opportunity cost no longer exists, as in thefuture, sudden price reductions as well as large increases in the value of the underlying assetmay be observed.


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7 CONCLUSION

This paper verified the robustness of the Grant, Vora & Weeks and Longstaff & Schwartz meth-ods when used to determine the incremental payoff of the carbon market in electricity generationprojects, considering that the CER price follows a jump-diffusion process. It was observed thatwhen the number of simulated price paths exceeds 20,000, and when the number of dates forearly exercise of the option exceeds 24, in this case being equivalent to monthly time periods,the value of the option as determined using the LSM method converged with the value esti-mated by the GVW method for all baseline scenarios. This analysis gains relevance, as in thepresence of jumps the assumption of no arbitrage is no longer valid and impedes the use of an-alytical solutions in the valuation of options, making the use of numerical methods necessaryin order to do so.

The particularities of the analyses carried out in this paper are due to the special characteristicsinvolved in the determination of the carbon market incremental payoff for grid-connected elec-tricity generation projects. In this case, in light of uncertainties associated with the dispatch ofpower plants within the system, for example, hydrological uncertainty, different project baselinescenarios may be observed in the future. In each of these scenarios it was noted that the baselinedoes not remain constant during the study period, but instead varies as the group of power plantsutilized within the grid varies.

This means that the relationship between the probability distributions of the CER price and thevalue of the option’s underlying asset may not be constant within its exercise period, eliminatingwhat is known as linearity between the stochastic processes of these two variables. As a conse-quence, results seen in the literature were not observed. The trigger curves for the underlyingasset of the considered management option did not present monotonicity, and hardly presentedthemselves as continuous, and in addition it was not possible to find a trigger curve that was validfor all baseline scenarios. These findings were considered original contributions of this paper.

Finally, different behavior was observed for the trigger curve of the derivative depending on thenature of the jumps considered in the simulation process of the CER price. When positive andnegative variations in value were considered, the trigger curves showed that during the first sixmonths of power plant construction the continuation value of the option was always superior toits immediate exercise value, indicating that such an option should not be exercise during thisperiod. This can be observed for the case study considered, as early exercise does not necessarilyimply an increase in revenue due to the sale of CERs. On the other hand, when only negativeprice variations were considered, the expectation that several CER price drops may be observedin the future reversed this logic.

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[1] BATISTA FRS, MELO ACG, TEIXEIRA JP & BAIDYA TKN. 2011a. The Carbon Market Incremen-

tal Payoff in Renewable Electricity Generation Projects in Brazil: a Real Options Approach. IEEE

Transaction on Power Systems, 26(3), August.

[2] BATISTA FRS, TEIXEIRA JP, BAIDYA TKN & MELO ACG. 2011b. Avaliacao dos Metodos deGrant, Vora & Weeks e dos Mınimos Quadrados na Determinacao do Valor Incremental do Mercado


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de Carbono nos Projetos de Geracao de Energia Eletrica no Brasil. Pesquisa Operacional, 31(1):

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http://www.centroclima.org.br/new2/ccpdf/cnae clima2.pdf.

[5] BRASIL. MINISTERIO DE MINAS E ENERGIA. 2006. Plano Decenal de Expansao de Energia Eletrica

2006-2015, Maio 2006. Available in: http://www.mme.gov.br.

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for Global Change, 8: 323–348.

[8] DIXIT A & PINDYCK RS. 1994. Investment under Uncertainty, Princeton University Press,

p. 394–405.

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The Journal of Finance Engineering, 5(3): 211–227.

[10] HULL J. 1993. Options, Futures and other Derivative Securities, Prentice-Hall International Inc.

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‘CLIMAFOR’ Approach to Estimate Baseline Carbon Emissions of a Forest Conservation Projectin the Selva Lacandona, Chiapas, Mexico. Mitigation and Adaptation Strategies for Global Change,

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Least-Squares Approach.The Review of Financial Studies, 14(1): 113–147.

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application of stochastic dual dynamic programming in official and agent studies in Brazil descriptionof the NEWAVE program. In: 16th Power Systems Computation Conference, Glasgow.

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nology Adoption Under Carbon Price Uncertainty: An Application to the Australian Electricity Gen-eration Sector. Economic Record, 82(S1): S64–S73.

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a hedge against uncertain gas and carbon prices? The Energy Journal, 27(4): 1. (print edition).

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[20] UNITED NATIONS FRAMEWORK CONVENTION ON CLIMATE CHANGE – UNFCCC. 1998. Kyoto

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APPENDIX

A Grant, Vora & Weeks Method for the valuation of financial options

The basic principle of this method consists in identifying the critical price of the underlying assetfor each time prior to its expiry, or in other words, the price at which the investor is indifferent asto whether to exercise the option or not at each point in time. Once these values are known, it canbe argued that the American derivative may be valuated in the same way as a European derivative,in other words, through the calculation of the arithmetic average of previously simulated values.

Note that from the critical option prices (S∗t ), two regions may be defined: the early exercise

region, where exercising would be the optimum decision, and the continuation region, where thebest strategy would be to wait until the next point in time to make a new decision. The indif-ference curve between these regions is called the Optimum Exercise Frontier, or the derivativeTrigger Curve.

Considering an American purchase option that may be exercised at any time t ∈ [0, T ], with anexercise price X , and an asset price at time t represented by St , according to the GVW methodthe value of this option (Ct ) can be determined according to equation 22:

Ct (St , X) = max{It , Ft } (22)

whereIt = max{St − X, 0} (23)

andFt = e−r.�t Et

[Ct+�t (St+�t , X)

]. (24)

In equation 22, note that the first term of the maximization operator represents the value ofimmediate exercise, while the second represents its continuation value. It is worth highlightingthat to determine the continuation value using the GVW method, previous knowledge of all ofthe critical prices between time t and the expiry of the option are required.

Since the critical price represents the price by which the immediate exercise value of the deriva-tive is equal to its continuation value, it is possible to define a boundary condition for S∗

t equatingequations 23 and 24:

S∗t − X = e−r.�t Et

[Ct+�t (S∗

t+�t , X)]. (25)


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From this equation, it can be concluded that the value of S∗ can easily be determined for the

derivative expiry date. Note that at maturity, the continuation value for the derivative is equal tozero, as there would not be any other opportunity for its exercise. Therefore, equation 25 can berewritten in the following way:

S∗T − X = 0 (26)

or, S∗T = X . Since the determination of S∗

t depends on the previous knowledge of all criticalprices at times before t , GVW propose that the trigger curvemay be determined recursively,using the Dynamic Programming technique.

The optimization process begins in the instant before expiry of the option, or at T − �t . Theholder of the purchase option can exercise it immediately or maintain the option “alive” until itsmaturity. Using equation 22, the value of the option may be determined in the following way:

CT −�t(ST−�t , X

) = max{

IT−�t , e−r.�t ET−�t[CT (ST , X)

]}.

The critical price (S∗T−�t) is identified by finding the value of ST−�t which satisfies condition

25. Assuming that it is possible to identify S∗T−�t , the optimization goes on to identify the value

of S∗T−2�t , which depends on the knowledge of values S∗

T−�t and S∗T . Using this logic, the

process continues until S∗0 is determined.

According to condition 25, determining the value of S∗t entails determining the continuation

value (F) associated with time t, however, information on future prices is still not known at thistime. Grant, Vora & Weeks solved this problem using the Monte Carlo Simulation technique

(MCS).

The MCS is initiated at time T − �t , adopting S∗T−�t = S∗

T as the initial condition. Oncean initial value for S∗

T−�t has been arbitrated, ST values are simulated in order to determinethe continuation value of the option. Where condition 25 is not satisfied, the value of S∗

T−�t

must be incremented and the MCS repeated. This routine must be carried out until condition25 is met.

The solution process continues recursively for the duration of the option’s lifetime. Once thetrigger curve of the derivative has been determined, the value of the option can be determined

through N Monte Carlo Simulations started at t0(t = 0). In this way, an initial price for theunderlying asset (S0) which can be observed in the market is considered. The early exercisehappens at the first instant in which the asset price surpasses the trigger curve. The final value of

the option is then determined using the average of the values obtained for each simulated path:

C0 = 1

N·

N∑w=1

e−r.τ · [ max(Swτ − X, 0

)]∀Swτ > S∗

τ . (27)

In this equation, τ represents the first instant at which the simulated price surpasses thetrigger curve.


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B Least Square Monte Carlo Method (LSM) for the valuation of financial options

As described in Appendix A, determining the immediate exercise value of an option may beconsidered to be a not so complicated task, however, a good estimate of its continuation value

is more difficult to obtain. As previously mentioned, in applying the GVW method this processrequires the performance of a large number of Monte Carlo Simulations, which can lead toincreased computational costs.

Considering this, Longstaff & Schwartz (2001) proposed a methodology that reduces the compu-

tational cost of the simulation methods. Compared to the GVW method, the main difference ofthe method proposed by Longstaff & Schwartz lies in the calculation of the continuation value.While GVW estimate this value using Monte Carlo Simulations, Longstaff & Schwartz propose

that regressions be performed using cross-sectional information about the financial asset. Thismethod is called the Least Square Monte Carlo Method, or LSM.

The first step of the LSM method consists in defining a finite number of dates on which earlyexercise of the option is possible. Therefore, considering T to be the expiry of the derivative,

it is assumed that the lifetime of the option can be divided into D intervals of equal size �t =T/D. Once N paths for the price of the underlying asset are simulated, Longstaff & Schwartzconsider that the option’s continuation price, at a given time t , can be can be initially set using

the following equation:

F(w, t) = EQ

[ T∑t j=t+�t

e−r(t j −t) · V (w, t j , t, T )/�t

], (28)

where t represents any time within a time interval [0, T ], w represents one of the simulated

paths, Q represents a measure of risk-neutral probabilities and �t represents all of the informationavailable at time t . Also note that in equation 28 V (w, t j , t, T ) represents the cashflow generatedby exercise of the option at any time t j > t . Since American options can be exercised only oncewithin each path w, it should be highlighted that there only exists one t j for which

V (w, t j , t, T ) > 0.

Despite equation 28 being used to obtain an initial estimate of the option’s continuation value ata given time t , Longstaff & Schwartz suppose that the continuation value F(w, t) may be better

estimated using cross-sectional regressions of the financial asset price. The algorithm is basedon the idea that F(w, t) can be represented using a combination of linear base functions (Bl ),whose constants are determined through a Least Squares Regression. This logic is represented byequation 29, where S represents the price of the underlying asset of the option, and al represents

the constant associated with each base function Bl .

F(w, t) =∞∑

l=0

al Bl (S) . (29)


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Equation 29 assumes infinite terms for the calculation of F(w, t), however, for practical rea-

sons this assumption is not computationally viable. In this case, the value of F(w, t) must beestimated using a number G < ∞ of base functions:

F(w, t) ≈ FG(w, t) =G∑

l=0

al Bl (S) . (30)

From equation 30 the LSM method estimates the value of FG(w, t) by regressing the continu-ation values initially calculated using equation 28 in relation to the predefined base functions.At a given time t , such a regression is performed considering only the paths in which the option

is found to be “in-the-money”, since it is only for these paths that the decision to exercise theoption early would be relevant.

Once the continuation value of the option (FG (w, t)) has been estimated for each path w, thedecision to exercise it early is made by comparing its immediate exercise value with the estimated

continuation value. Just as is seen in the GVW method, the iterative process of the LSM methodis recursive. The value of the option (CL S) is approximated by calculating the arithmetic averageof the sum of all cashflows V (w, t j , t, T ) where exercise of the option would be optimum, or:

CL S = 1

N

N∑w=1

T∑t j=�t

e−r.t j V (w, t j , 0, T ) . (31)

It remains clear that the relatively low computational cost associated with the LSM method

represents its main advantage when compared with other methods involving MCS in deriva-tive valuation. In addition, such as can be observed for the GVW method, this method may alsopermit the valuation of different types of options involving different stochastic processes, or evenoptions with different dimensions.


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